Good luck with your searching, I think its going to be a challenging one! I think you have reasonable grounds to believe Louis and Lewis are the same person as there is a birth registration for Lewis of the right date but No records at all for a Louis, so he must have changed the spelling.
I tried to find the 1891 census for you however the family are not there. I suspect they moved to England from Russia had children Lewis and Betsy, then moved back to Russia for some reason (a little unusual for a Jewish family of this time) where they had daughter Yetta, then moved back to england in 1891 where next son Jacob was born but this was after the census was taken!
I have no idea how you would go about sourcing Russian records though, apologies.